2026/07/18 by Hongzhang Chen, Yongtao Li, Quanyu Tang
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In this paper, we use the spectral surplus λ(G) - √(m) to measure how far G lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph G with m≥ 3 edges and λ(G) ≥ 1 + √(m-2) contains at least m-2 triangles, with equality if and only if G = K3 \vee \tfracm-33 K1. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer t(G) ≥ \lfloor \tfrac12(√(m)-1) \rfloor proved by Ning and Zhai, and the second layer t(G) ≥ \tfracm-12 by Zhang and Zhai. (b) Every m-edge graph G satisfies t(G) ≥ m(λ- √(m) ), with equality if and only if G is complete bipartite. Consequently, λ(G) ≥ √(m) + q forces t(G) > q m for every real q > 0. This is an edge-spectral counterpart of the Lovász--Simonovits theorem, and it improves the Bollobás--Nikiforov bound t(G) ≥ \tfrac13 λ(λ2 - m) in the range √ m ≤ λ(G) ≤ 1.3√ m . (c) Every m-edge Nosal graph G contains a book of size greater than \tfrac14 √(m). This improves two recent results on the booksize constant: \tfrac124 proved by Li, Liu and Zhang, and \tfrac19 by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant \tfrac13. (d) Every m-edge Nosal graph G contains at least (\tfrac18 - o(1)) m copies of the kite C4+=B2, and the constant \tfrac18 is best possible. This determines the sharp asymptotic constant for counting C4+ and strengthens the Ω(m) bound of Li, Liu and Zhang.